Computing class polynomials for abelian surfaces - Cryptology

28 Pages·2013·0.24 MB·English

Computing class polynomials for abelian surfaces
Andreas Enge* and Emmanuel Thomé†
17 May 2013
Abstract
Wedescribeaquasi-linearalgorithmforcomputingIgusaclasspolynomialsofJa-
cobians of genus 2 curves via complex floating-point approximations of their roots.
After providing an explicit treatment of the computations in quartic CM fields
and their Galois closures, we pursue an approach due to Dupont for evaluating #-
constants in quasi-linear time using Newton iterations on the Borchardt mean. We
report on experiments with our implementation and present an example with class
number 17608.
1 Introduction
Igusa class polynomials describe the complex multiplication points in the moduli space
of principally polarised abelian surfaces, that is, they parameterise abelian varieties of
dimension2withcomplexmultiplicationbyamaximalorderofaquarticCMfield. Such
abelian surfaces are Jacobians of hyperelliptic curves of genus 2, so that by computing
Igusa class polynomials one may obtain genus 2 curves over finite fields with known
Jacobian cardinality.
In the dimension 1 case of elliptic curves, several approaches have been described in
the literature. While the output of the algorithms (a large polynomial) is of exponential
size in the input (a number field described by a single integer), all of these approaches
may lead to an algorithm with a complexity that is quasi-linear (up to logarithmic fac-
tors)initsoutputsize: Thecomplexanalyticmethodusesfloatingpointapproximations
to the roots of the class polynomials [16]; the p-adic approach starts from a curve with
the given endomorphism ring over a small finite field and lifts its invariants to a p-adic
field[13,8]; theChineseremainderingapproachcombinescurvesoverseveralsmallprime
fields [2].
*INRIA, LFANT, F-33400 Talence, France
CNRS, IMB, UMR 5251, F-33400 Talence, France
Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France
[e-mail]
†INRIA, CARAMEL, Nancy, France
[e-mail]
1
In principle, the same approaches apply to abelian surfaces. A 2-adic algorithm is
described in [21], and there are currently attempts at making the Chinese remainder
based method more efficient [28]. So far, 2-adic lifting appears to have been the most
successful approach: The Echidna database maintained by Kohel1 contains Igusa class
polynomials, the largest of which is of degree 576 and has been obtained by lifting from
a curve over F 2.
26
A detailed description of the complex analytic approach, together with complexity
analyses of its different steps, has recently been given in [33, 34]. Our work pushes
the limits for the attainable degrees of Igusa class polynomials: We present an example
of degree 17608. Moreover, relatively small class polynomials (say, below degree 150)
can be computed in matters of seconds. The key tool in this approach is the use of a
quasi-linear algorithm for the computation of #-constants, initially described in [15].
This article is organised as follows. §2 presents the necessary background material
for discussing the complex multiplication theory of abelian surfaces and states the gen-
eral algorithm. §§3 and 4 show how to explicitly (providing concrete descriptions for the
occurring number fields, maps between them and their embeddings) and symbolically
compute an appropriate set of reduced period matrices, which form the input of the
computationally expensive step of computing #-constants, detailed in §5. The recogni-
tion as algebraic numbers of the coefficients of the Igusa class polynomials from their
approximations by complex embeddings is described in §6, and experimental results are
given in §§7 and 8.
Allcomputationspresentedinthisarticlehavebeenachievedwiththesoftwarepack-
age Cmh[20], released under the GNU General Public License.
2 Complex multiplication theory
In this section, we provide a concise introduction to the theory of complex multipli-
cation of principally polarised abelian surfaces or, equivalently, Jacobians of genus 2
hyperelliptic curves over the complex numbers, to the extent needed to describe our
algorithms and implementation. The presentation follows [34], and proofs are given in
[31, 30, 34, 33].
2.1 Quartic CM fields and abelian surfaces
A CM field K is an imaginary-quadratic extension of a totally real number field K .
0
We denote by (cid:20) indiscriminately the complex conjugation on C and the automorphism
generating Gal(K/K ). For any embedding φ : K ! C, we have (cid:20)(cid:14)φ = φ(cid:14)(cid:20), which
0
justifies the notation φ = (cid:20)(cid:14)φ.
Quartic CM fields K of degree 4 over Q come in three Galois types. Generically,
K/QisnotGalois, theGaloisclosureL/K isofdegree2, andGal(L/Q)isisomorphicto
the dihedral group D . L is itself a CM field, and the complex conjugation of L, which
4
1http://echidna.maths.usyd.edu.au/echidna/dbs/complex_multiplication2.html
2Personal communication
2
we denote again by (cid:20), restricts to the complex conjugation of K. If K/Q is Galois,
it may be either cyclic or biquadratic. We will not consider the biquadratic case in
the following, since then the abelian surfaces of which it is the endomorphism algebra
are products of elliptic curves; so from now on, all Galois quartic CM fields are tacitly
understood to be cyclic.
A CM type of a quartic CM field K is a set (cid:8) = fφ ;φ g of two embeddings K !
1 2
C such that φ ̸= φ ; that is, it contains one out of each pair of complex-conjugate
2 1
′
embeddings. Two CM types (cid:8) and (cid:8) are equivalent if there is an automorphism (cid:27) of K
such that (cid:8)′ = (cid:8)(cid:14)(cid:27); in particular, (cid:8) and (cid:8) are equivalent. If K/Q is Galois, there
is only one equivalence class of CM types; otherwise, there are two inequivalent classes
(cid:8) = fφ ;φ g and (cid:8)′ = fφ ;φ g.
1 2 1 2
For a given CM type (cid:8) = fφ(1;φ2g, its reflex field is th)e field Kr generated over Q by
the type traces, that is, Kr = Q fφ (x)+φ (x) : x 2 Kg ; it is itself a quartic CM field
1 2
and we denote by Kr its real-quadratic subfield. Equivalent CM types yield conjugate
0
reflex fields. In the Galois case, K and Kr are isomorphic, while in the dihedral case,
they are not isomorphic, but the two reflex fields for the two inequivalent CM types are.
In both cases, there is a natural way of defining a dual CM type (cid:8)r = fφr;φrg of Kr,
1 2
andthereflexfieldofKr isisomorphictoK. Definethe(dual)type norm N : Kr ! K
(cid:8)r
by x 7! φr(x)φr(x), so that
1 2
N N = N; (1)
(cid:8)r (cid:8)r
this map extends to ideals and ideal classes.
In §3, we provide explicit equations for all occurring number fields and consider their
embeddings from an effective point of view.
Let a be a fractional ideal of O . A CM type (cid:8) = fφ ;φ g induces an embedding
K 1 2
K ! C2, x 7! (φ (x);φ (x)), under which (cid:8)(a) is a lattice of rank 4. Its cokernel
1 2
C2/(cid:8)(a), a complex torus of genus 2, is an abelian surface. Let (cid:14)(cid:0)1 = fy 2 K : Tr(xy) 2
K
Z 8x 2 O g be the codifferent ideal of K. Assume that (aa(cid:14) )(cid:0)1 is principal and
K K
generated by some (cid:24) 2 K such that φ ((cid:24));φ ((cid:24)) 2 iR>0; in particular, (cid:24)(cid:24) 2 K is totally
1 2 0
negative. Then E : (cid:8)(K)2 ! Q;((cid:8)(x);(cid:8)(y)) 7! Tr((cid:24)xy) is a symplectic form over Q
(cid:8);(cid:24)
whichtakesintegralvalueson(cid:8)(a)2. BytensoringwithR,oneobtainsasymplecticform
C2 ! R such that (x;y) 7! E (ix;y) is symmetric and positive definite, a principal
(cid:8);(cid:24)
polarisation on C2/(cid:8)(a). ( )
The principally polarised abelian surface A((cid:8);a;(cid:24)) = C2/(cid:8)(a);E has complex
(cid:8);(cid:24)
multiplication by O ; conversely, any such surface can be obtained up to isomorphism
K
′ ′ ′
in this way. Two principally polarised abelian surfaces A((cid:8);a;(cid:24)) and A((cid:8);a;(cid:24) ) are
isomorphic if and only if (cid:8) = (cid:8)′ (up to equivalence) and there is a u 2 K(cid:3) such that
a′ = ua and (cid:24)′ = (uu)(cid:0)1(cid:24). In particular this implies that uu 2 K is totally positive,
0
and that we may assume a to be an integral ideal of O .
K
2.2 The Shimura group, its type norm subgroup and cosets
The Igusa invariants to be defined in §2.3 determine the moduli space M of principally
polarised complex abelian surfaces, which has a model over Q. Let M be the subset
K;(cid:8)
3
of surfaces A((cid:8);a;(cid:24)) obtained from an integral ideal of O and the CM type (cid:8) as
K
described in §2.1. Then M is stable under Gal(Q/Kr). If K is cyclic, then M is
K;(cid:8) 0 K;(cid:8)
even stable under Gal(Q/Q). Otherwise let (cid:8)′ be inequivalent with (cid:8). Then M and
K;(cid:8)
MK;(cid:8)′ are disjoint and conjugate under Gal(K0r/Q) [34, Lemmata 1.1 and 2.1].
Let the Shimura class group C be defined by
{
C = (a;u) : a a fractional ideal of O ;aa = uO ; and u 2 K totally positiveg/ (cid:24)
K K 0
(2)
with component-wise multiplication. The equivalence relation denoted (cid:24) above is the
one induced by principal ideals, more precisely the equivalence modulo the subgroup
given by the (vO ;vv) with v 2 K(cid:3) and vv 2 K totally positive.
K 0
By the discussion of §2.1, C acts regularly on M via
K;(cid:8)
(b;u)(cid:1)A((cid:8);a;(cid:24)) = A((cid:8);b(cid:0)1a;u(cid:24)): (3)
Consider the dual type norm map N : Cl ! C;b 7! (N (b);N(b)); which is well
(cid:8)r Kr (cid:8)r
defined by (1). For any A((cid:8);a;(cid:24)), the action induced by N (Cl ) is that of the Galois
(cid:8)r Kr
group of the field of moduli of A((cid:8);a;(cid:24)) over Kr [34, Theorem 9.1]; otherwise said, the
field of moduli is the fixed field of ker(N ) inside the Hilbert class field of Kr. The
(cid:8)r
cokernel of N is elementary abelian of exponent 1 or 2 [34, Theorem 2.2], so M
(cid:8)r K;(cid:8)
splits into orbits under C of size jim(N )j, and the number of orbits is a power of 2.
(cid:8)r
As stated above, these orbits are in fact defined over Kr, with the orbits of M and
0 K;(cid:8)
MK;(cid:8)′ being mapped to each other by Gal(K0r/Q).
2.3 #-functions, Igusa invariants and class polynomials
Given an ideal a and a principal polarisation E as in §2.1, one may choose a Z-basis
(cid:8);(cid:24)
((cid:11)1;(cid:11)2;(cid:11)3;(cid:11)4) of a such that v1 = (cid:8)((cid:11)1), v2(= (cid:8)((cid:11)2),)w1 = (cid:8)((cid:11)3), w2 = (cid:8)((cid:11)4) form
0 id
a symplectic basis, for which E becomes 2 . That the change of basis is
(cid:8);(cid:24) (cid:0)id 0
2
defined over Z and not only over R follows f(rom th)e princip(ality of)the polarisation; we
also call this basis of a symplectic. Let V = v v , W = w w 2 C2(cid:2)2. Rewriting
1 2 1 2
the amb(ient vector)space C2 and (cid:8)(a) in the basis spanned by w1 and w2, we obtain
(cid:8)(a) = Ω id Z4 with the period matrix
(cid:8);a;(cid:24) 2
Ω = W(cid:0)1V (4)
(cid:8);a;(cid:24)
{ }
intheSiegelhalfspaceH = Ω 2 C2(cid:2)2 : Ω symmetric and ℑ(Ω) positive definite . The
2
symplectic group Sp (Z) acts on H by
4 2
( )
A B
Ω = (AΩ+B)(CΩ+D)(cid:0)1;
C D
where A, B, C, D 2 Z2(cid:2)2. As in the case of genus 1, a fundamental domain for H
2
exists under the action of Sp (Z). Reduction into the fundamental domain is discussed
4
in §3.3.
4
( )
a
#-constants are certain modular forms of weight 1/2 for Sp (Z). Let a = 1 ,
4 a
( ) 2
( )
b = b1 2 1Z 2 be two vectors of #-characteristics. Then for Ω 2 H ,
b 2 2
2
∑
#16a1+8a2+4b1+2b2(Ω) = #a;b(Ω) = e2(cid:25)i(12(n+a)⊺Ω(n+a)+(n+a)⊺b): (5)
n2Z2
Only the even #-constants # for i 2 T = f0;1;2;3;4;6;8;9;12;15g are not identically 0.
i
The following duplication formulæ relate the values of the squares of the ten even
#-constants in the argument Ω with the values of the four fundamental #-constants
# ;:::;# (which have a = 0) in the argument Ω/2 (omitted from the formulæ for the
0 3
sake of conciseness).
4#2(Ω) = #2+#2+#2+#2 4#2(Ω) = 2# # (cid:0)2# #
0 0 1 2 3 6 0 2 1 3
4#2(Ω) = 2# # +2# # 4#2(Ω) = #2+#2(cid:0)#2(cid:0)#2
1 0 1 2 3 8 0 1 2 3
4#2(Ω) = 2# # +2# # 4#2(Ω) = 2# # (cid:0)2# # (6)
2 0 2 1 3 9 0 1 2 3
4#2(Ω) = 2# # +2# # 4#2 (Ω) = #2(cid:0)#2(cid:0)#2+#2
3 0 3 1 2 12 0 1 2 3
4#2(Ω) = #2(cid:0)#2+#2(cid:0)#2 4#2 (Ω) = 2# # (cid:0)2# #
4 0 1 2 3 15 0 3 1 2
Denote by h the following modular forms of weight j:
j
∑ ∑
h = #8; h = (cid:6)(# # # )4;
4 i 6 i j k
i2T 15triples(i;j;k)2T3
∏ ∑ (7)
h = #2; h = (# # # # # # )4;
10 i 12 i j k l m n
i2T 15tuples(i;j;k;l;m;n)2T6
for the exact definitions, see [34, §II.7.1]. These generate the ring of holomorphic Siegel
modular forms over C, see [25, Corollary p. 195] and [34, Remark 7.2]. The moduli
space of principally polarised abelian surfaces is of dimension 3 and p[arameterised b]y
absolute Igusa invariants, modular functions (thus of weight 0) in Z h ;h ;h ;h(cid:0)1 .
4 6 12 10
Different sets of invariants have been suggested in the literature. The most cited one is
Spallek’s, who uses a system in the linear span of h512, h312h4, h212h6 [32, Satz 5.2]. Streng
h6 h4 h3
10 10 10
defines invariants with the minimal powers of h in the denominator as
10
h h h2h h5
j = 4 6; j = 4 12; j = 4 : (8)
1 h 2 h2 3 h2
10 10 10
The principally polarised abelian surfaces A((cid:8);a;(cid:24)) are parameterised by the triples
of singular values (j (Ω);j (Ω);j (Ω)) in the period matrices Ω = Ω , which may
1 2 3 (cid:8);a;(cid:24)
be obtained from the action of the Shimura class group C on a fixed base point (cid:12) =
((cid:8);a ;(cid:24) ). The singular values lie in the subfield of the Hilbert class field of Kr
(cid:8) (cid:8)
given in §2.2. Following the discussion there, the Igusa class polynomials I (X) =
i
5
∏
(X (cid:0)j (Ω )) are defined over Q. More precisely their irreducible factors,
((cid:8);a;(cid:24)) i (cid:8);a;(cid:24)
over Kr in the dihedral case or Q in the cyclic case, are given by
0
∏ ( )
X (cid:0)ji(ΩCC′(cid:1)(cid:12)) ,
C2N(cid:8)r(ClKr)
where (cid:8) is one CM type and C′ 2 C/N (Cl ).
(cid:8)r Kr
In the following, we fix a CM type (cid:8) (for its explicit description, see §3) and a base
point (cid:12) = ((cid:8);a ;(cid:24) ) and let
(cid:8) (cid:8)
∏
H1(X) = (X (cid:0)j1(ΩC(cid:1)(cid:12))): (9)
C2N(cid:8)r(ClKr)
As elements of the same class field, the singular values of j and j are rational ex-
2 3
pressions in the singular value of j . Computationally, it is preferable to use the Hecke
1
representation in the trace-dual basis to keep denominators small. We thus define poly-
nomials H^ and H^ through j H′(j ) = H^ (j ) with
2 3 i 1 1 i 1
∑ ∏
H^i(X) = ji(ΩC(cid:1)(cid:12)) (X (cid:0)j1(ΩD(cid:1)(cid:12))) (10)
C2N(cid:8)r(ClKr) D2N(cid:8)r(ClKr)nfCg
for i 2 f2;3g, where H , H^ , H^ 2 Kr[X] in the dihedral case and 2 Q[X] in the cyclic
1 2 3 0
case.
2.4 Algorithm for Igusa class polynomials
We briefly summarise the algorithm for computing class polynomials.
Algorithm 1
Input: CM field K and CM type (cid:8) = fφ ;φ g of K
1 2
Output: Irreducible class polynomials H , H^ , H^ 2 Kr[X] in the dihedral case and
1 2 3 0
2 Q[X] in the Galois case
1) Compute N (Cl ) = f(b ;u );:::;(b ;u )g (cid:18) C.
(cid:8)r Kr 1 1 h h
{
(a a (cid:14) )(cid:0)1 = ((cid:24) );
2) Compute a base point (cid:12) = ((cid:8);a ;(cid:24) ) such that (cid:8) (cid:8) K (cid:8)
(cid:8) (cid:8) φ ((cid:24) );φ ((cid:24) ) 2 iR>0:
1 (cid:8) 2 (cid:8)
3) Enumerate fC(cid:1)(cid:12) = ((cid:8);b(cid:0)1a ;u (cid:24) ); C = (b ;u ) 2 N (Cl )g and compute the
i (cid:8) i (cid:8) i i (cid:8)r Kr
associated period matrices Ωi = ΩC(cid:1)(cid:12) for i = 1;:::;h.
4) For i = 1;:::;h, compute the fundamental #-constants # (Ω /2);:::;# (Ω /2);
0 i 3 i
then deduce the squares of the ten even(#-constants #2k(Ωi) )by (6), the values
h (Ω ) by (7) and finally the triples J = j (Ω );j (Ω );j (Ω ) by (8).
k i i 1 i 2 i 3 i
∏ ∑ ∏
5) Let H = h (X (cid:0)J ), H^ = h J (X (cid:0)J ) 2 C[X] for k 2 f2;3g.
1 i=1 i;1 k i=1 i;k l̸=i l;1
6
6) Recognise the coefficients of H , H^ , H^ as elements of Kr or Q, respectively.
1 2 3 0
The different steps of the algorithm and our implementation are detailed in the
following chapters. The symbolic computations related to number fields in Steps 1)
and 2) and to the period matrices Ω in Step 3) are described in §3. Step 1) is treated
i
in §4.2, Step 4) in §5 and Step 6) in §6.
3 Explicit equations and symbolic period matrices
While Algorithm 1 in fine works with complex approximations obtained via CM types,
it starts from an algebraic setting. In this section, we examine how to carry out the
computations as far as possible symbolically with algebraic numbers, which relieves us
from the need to decide on the necessary precision early on. In particular, in §3.1 we
replace the complex embeddings forming a CM type by algebraic embeddings into the
compositum L of all involved fields, followed by a “universal” embedding of L into C.
Taking preimages under , the entries of the period matrices Ω 2 C2(cid:2)2 may then be
interpreted as elements of the reflex field and may be handled symbolically. We then fix
a model for the CM field K in §3.2 and derive explicit equations for all considered fields
and embeddings.
Recall the notation of §2: K is a quartic CM field, K its real quadratic subfield and
0
L its Galois closure with Galois group G. We consider only the dihedral case [L : K] = 2
and G = D and the cyclic case L = K and G = C . Let (cid:8) = (φ ;φ ) be a CM type,
4 4 1 2
where φ , φ : K ! C are two complex embeddings of K with φ ̸= φ , and let Kr be
1 2 2 1
the reflex field of K with respect to (cid:8).
3.1 Galois theory, embeddings and period matrices
3.1.1 The dihedral case
Galois theory. Let K = Q(y) be a non-Galois quartic CM field. The following state-
ments are easily seen to be true when choosing a generator y such that z = y2 belongs to
the real subfield K , so that K = Q[Y]/(Y4+AY2+B) for some A;B 2 Q. The Galois
0
closure of K is then L = K(y′) = Q(y;y′), where the roots of the minimal polynomial
of y in L are (cid:6)y and (cid:6)y′ (the former could be identified with φ (y) in (12), the latter
1
with φ (y)). The automorphisms in G = Gal(L/Q) are uniquely determined by their
2
′
images on y and y , and we obtain the following diagram of fields and Galois groups:
L = KKr
⟨(cid:26)⟩ ⟨(cid:27)⟩
⟨(cid:20)⟩
K (cid:3) Kr
⟨(cid:20)j ⟩ ⟨(cid:20)j ⟩
K Kr
K Kr
0 0
Q
.............
7
Here the automorphisms are given by
(cid:26) : y 7! y; y′ 7! (cid:0)y′ of order 2
(cid:27) : y 7! y′; y′ 7! y of order 2, which fixes the generator y+y′ of Kr
(cid:28) : y 7! y′; y′ 7! (cid:0)y of order 4
(cid:20) = (cid:28)2 : y 7! (cid:0)y; y′ 7! (cid:0)y is the complex conjugation:
So G is the dihedral group D with generators (cid:28) of order 4 and (cid:26) (or (cid:27)) of order 2 and
4
additional relation (cid:26)(cid:28)(cid:26) = (cid:28)3, and with ⟨(cid:20)⟩ as its centre.
Embeddings and CM types. There is a unique embedding : L ! C such that
φ = j and φ = ( (cid:27))j (where multiplication denotes composition), which can be
1 K 2 K
seen as follows. First of all, there are two embeddings which, restricted to K, yield
φ ; we denote them by and ′ = (cid:26). Now there is s 2 G, uniquely defined up to
1 1 1 1
multiplication by (cid:26) from the right, such that φ = ( s)j . Since φ ̸= φ and φ ̸= φ ,
2 1 K 2 1 2 1
the automorphism s is neither 1, (cid:26), (cid:20) = (cid:28)2 nor (cid:20)(cid:26) = (cid:28)2(cid:26). This leaves s as one of
(cid:28) = (cid:26)(cid:27), (cid:28)(cid:26) = (cid:26)(cid:27)(cid:26), (cid:28)3 = (cid:27)(cid:26) or (cid:28)3(cid:26) = (cid:27). If sj = (cid:27)j = ((cid:27)(cid:26))j , we may choose = .
K K K 1
Otherwise, sj = (cid:26)(cid:27), and ( ′(cid:27))j = ( (cid:26)(cid:27))j = ( s)j = φ , so we choose = ′.
K 1 K 1 K 1 K 2 1
Period matrices. Let ((cid:11) ;:::;(cid:11) ) be a symplectic basis for the ideal a of K with
1 4
respect to E as defined in §2.3. Then
(cid:8);(cid:24)
( ) (( ))
φ ((cid:11) ) φ ((cid:11) ) (cid:11) (cid:11)
V = 1 1 1 2 = 1 2 ;
φ ((cid:11) ) φ ((cid:11) ) (cid:11)(cid:27) (cid:11)(cid:27)
( 2 1 2 2 ) (( 1 2))
φ ((cid:11) ) φ ((cid:11) ) (cid:11) (cid:11)
W = 1 3 1 4 = 3 4
φ ((cid:11) ) φ ((cid:11) ) (cid:11)(cid:27) (cid:11)(cid:27)
2 3 2 4 3 4
and
( )
1 (cid:11) (cid:11)(cid:27) (cid:0)(cid:11) (cid:11)(cid:27) (cid:11) (cid:11)(cid:27) (cid:0)(cid:11) (cid:11)(cid:27)
Ω = W(cid:0)1V = (M) with M = 4 1 1 4 4 2 2 4
(cid:8);a;(cid:24) (cid:11) (cid:11)(cid:27) (cid:0)(cid:11) (cid:11)(cid:27) (cid:11) (cid:11)(cid:27) (cid:0)(cid:11) (cid:11)(cid:27) (cid:11) (cid:11)(cid:27) (cid:0)(cid:11) (cid:11)(cid:27)
3 4 4 3 3 1 1 3 3 2 3 2
(11)
by (4). The entries of M are invariant under (cid:27) and thus elements of Kr.
Remark. It is crucial to choose out of the two embeddings : L ! C that extend
φ the one compatible with φ . The other one corresponds to the second CM type
1 2
(cid:8)′ = (φ ;φ ) with reflex field (Kr)′ and Gal(L/(Kr)′) = ⟨(cid:20)(cid:27)⟩ = ⟨(cid:26)(cid:27)(cid:26)⟩.
1 2
3.1.2 The cyclic case
Here we have the much simpler situation
8
K
⟨(cid:20)⟩ = ⟨(cid:27)2⟩
K
0
Q.....
We may choose = φ . Then there is a uniquely determined (cid:27) 2 Gal(K/Q) such that
1
φ = φ (cid:27), and trivially M of (11) has entries in Kr. In general, they will not lie in a
2 1
subfield: Since (cid:27) is neither the identity nor complex conjugation, it is of order 4.
3.2 Number field computations
InthissectionweshowhowtoexpresstheelementsofthereflexfieldKr andthenormal
closureLinconsistentways,soastobeabletocomputetypenormsandentriesofperiod
matrices as given by (4). We use the same notation for elements of the Galois group G
of L/Q as in §3.1.
3.2.1 The dihedral case
Field equations. By choosing generating elements as in §3.1.1 we may assume that
( )
K = Q(z) = Q[Z]/ Z2+AZ +B with A;B 2 Z>0;A2(cid:0)4B > 0;
0 ( )
K = Q(y) = Q[Y]/ Y4+AY2+B :
We then select the CM type (cid:8) = (φ ;φ ) with
1 2
√ √
p p
A+ A2(cid:0)4B A(cid:0) A2(cid:0)4B
φ (y) = i ; φ (y) = i ; (12)
1 2
2 2
′
where all the real roots are taken to be positive; the other CM type is (cid:8) = (φ ;φ ) with
1 2
φ (y) = (cid:0)φ (y). Recall from §3.1.1 the notations Gal(L/K) = ⟨(cid:26)⟩, Gal(L/Kr) = ⟨(cid:27)⟩,
2 2
and let : L ! C be such that φ = j and φ = ( (cid:27))j . The reflex field Kr is
1 K 2 K
generated by the type traces of K; letting yr = y+y(cid:27), the equality
(yr) = (y)+( (cid:27))(y) = φ (y)+φ (y) (13)
1 2
shows that we may consider yr as a generator of Kr. This gives the equations
( )
Kr = Q(zr) = Q[Zr]/ (Zr)2+ArZr +Br with Ar = 2A;Br = A2(cid:0)4B;
0 ( )
Kr = Q(yr) = Q[Yr]/ (Yr)4+Ar(Yr)2+Br :
The minimal polynomials of yr over K and y over Kr follow:
(yr)2(cid:0)2yyr +(2y2+A); (y)2(cid:0)yry+((yr)2+A)/2:
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We write the Galois closure L = KKr as the compositum generated over K or Kr by
t = y+yr. The minimal polynomial of t is the resultant
( )
h(T) = Res Y4+AY2+B;(T (cid:0)Y)2(cid:0)2Y(T (cid:0)Y)+(2Y2+A)
Y ( )
= Res (Yr)4+Ar(Yr)2+Br;(T (cid:0)yr)2(cid:0)yr(T (cid:0)yr)+((yr)2+A)/2
Yr
= T8+10AT6+(33A2(cid:0)14B)T4+(40A3(cid:0)70AB)T2+16A4(cid:0)200A2B+625B2:
Conversions and Galois actions. Weareinterestedintheactionof(cid:26), thegenerator
of Gal(L/K), on Kr, and in the action of (cid:27), the generator of Gal(L/Kr), on K. The
defining equations give:
yr +(yr)(cid:26) = 2y; yr(yr)(cid:26) = yr(yr)(cid:26) = 2y2+A; y(cid:26) = y;
( )
y+y(cid:27) = yr; yy(cid:27) = (yr)2+A /2; (yr)(cid:27) = yr:
An element of K is converted to an element of L, as a relative extension of Kr, using
the identity y = t(cid:0)yr; in the opposite direction we use yr = t(cid:0)y. The entries of the
matrix M of (11) are obtained from elements of K and their images under (cid:27), and need
to be expressed as elements of Kr. For this we use the identity y(cid:27) = yr(cid:0)y. This allows
to work in the relative extension L/Kr and to easily identify elements of Kr.
Dual type norms. For an ideal b of Kr, we have
N (b) = N (bO );
(cid:8)r L/K L
see [9, §3.1]. Computing dual type norms thus reduces to conversions in relative exten-
sions as described above.
3.2.2 The cyclic case
We may use the same type of equations for K and K as in the dihedral case, and may
0
fix = φ as in (12). Fixing an arbitrary element (cid:27) 2 Gal(K/Q) of order 4, we obtain
1
φ = φ (cid:27). Then the dual type norm for an ideal b of K is computed as
2 1
(cid:27)
N (b) = bb ;
(cid:8)r
see [9, §3.1].
3.3 Symbolic reduction of period matrices
Gottschling in [22] has determined a finite set of inequalities describing a fundamental
domain F2 for Sp4(Z)nH2, which directly translate into an algorithm for reducing an
element of H into F . As the Igusa functions introduced in §2.3 are modular for
2 2
Sp (Z), we may transform all period matrices occurring in Algorithm 1 into F . A
4 2
period matrix Ω is reduced if ℜ(Ω) has coefficients between (cid:0)1 and 1 (which may be
2 2
obtainedbyreducingmoduloZ),ifthebinaryquadraticformdefinedbyℑ(Ω)isreduced
10